A note on the cyclical edge-connectivity of fullerene graphs

A fullerene graph is a 3-regular (cubic) and 3-connected spherical graph that has exactly 12 pentagonal faces and other hexagonal faces. The cyclical edge-connectivity of a graph G is the maximum integer k such that G cannot be separated into two components, each containing a cycle, by deletion of fewer than k edges. Došlić proved that the cyclical edge-connectivity of every fullerene graph is equal to 5. By using Euler’s formula, we give a simplified proof, mending a small oversight in Došlić’s proof. Further, it is proved that the cyclical connectivity of every fullerene graph is also equal to 5.     

On decompositions of leapfrog fullerenes

It is shown that given a fullerene F with the number of vertices n divisible by 4, and such that no two pentagons in F share an edge, the corresponding leapfrog fullerene Le(F) contains a long cycle of length 3n − 6 missing out only one hexagon.     

New Lower Bound on the Number of Perfect Matchings in Fullerene Graphs

A fullerene graph is a cubic and 3-connected plane graph (or spherical map) that has exactly 12 faces of size 5 and other faces of size 6, which can be regarded as the molecular graph of a fullerene. T. Doli [3] obtained that a fullerene graph with p vertices has at least (p+2)/2 perfect matchings by applying the recently developed decomposition techniques in matching theory of graphs. This note gets a better lower bound 3(p+2)/4 of the number of perfect matchings of a fullerene graph by finding its 2-extendability. This property further implies a chemical consequence that every derivative of a fullerene by substituting any two pairs of adjacent carbon atoms permits a Kekulé structure.     

Cyclic edge-cuts in fullerene graphs

In this paper, we study cyclic edge-cuts in fullerene graphs. First, we show that the cyclic edge-cuts of a fullerene graph can be constructed from its trivial cyclic 5- and 6-edge-cuts using three basic operations. This result immediatelly implies the fact that fullerene graphs are cyclically 5-edge-connected. Next, we characterize a class of nanotubes as the only fullerene graphs with non-trivial cyclic 5-edge-cuts. A similar result is also given for cyclic 6-edge-cuts of fullerene graphs.     

Numbers of faces in disordered patches

It has been shown that the boundary structure of patches with all faces of the same size k, all interior vertices of the same degree m and all boundary vertices of degree at most m determines the number of faces of the patch (Brinkmann et al., Graphs and discovery, 2005; Guo et al., Discrete Appl Math 118(3):209–222, 2002). In case of at least two defective faces, that is faces with degree k′ ≠ k, it is well known that this is not the case. The most famous example for this is the Endo–Kroto C ₂-insertion (Endo and Kroto, J Phys Chem 96:6941–6944, 1992). Patches with alimited amount of disorder are especially interesting for the case k = 6, m = 3 and k′ = 5. This case corresponds to polycyclic hydrocarbons with a limited number of pentagons and to subgraphs of fullerenes. The last open question was the case of exactly one defective face or vertex. In this paper we generalize the results of Brinkmann et al. (2005) and Guo et al. (2002) and in some cases corresponding to Euclidean lattices also deal with patches that have vertices of degree larger than m on the boundary, have sequences of degrees on the boundary that are identical only modulo m and have vertex and face degrees in the interior that are multiples of m, resp. k. Furthermore we prove that in case of at most one defective face with a degree that is not a multiple of k the number of faces of a patch is determined by the boundary. This result implies that fullerenes cannot grow by replacing patches of a restricted size.     

Fullerene graphs have exponentially many perfect matchings

A fullerene graph is a planar cubic 3-connected graph with only pentagonal and hexagonal faces. We show that fullerene graphs have exponentially many perfect matchings.     

An Upper Bound for the Clar Number of Fullerene Graphs

A fullerene graph is a three-regular and three-connected plane graph exactly 12 faces of which are pentagons and the remaining faces are hexagons. Let F _n be a fullerene graph with n vertices. The Clar number c(F _n ) of F _n is the maximum size of sextet patterns, the sets of disjoint hexagons which are all M-alternating for a perfect matching (or Kekulé structure) M of F _n . A sharp upper bound of Clar number for any fullerene graphs is obtained in this article: . Two famous members of fullerenes C₆₀ (Buckministerfullerene) and C₇₀ achieve this upper bound. There exist infinitely many fullerene graphs achieving this upper bound among zigzag and armchair carbon nanotubes.     

On the anti-Kekulé number of leapfrog fullerenes

The Anti-Kekulé number of a connected graph G is the smallest number of edges that have to be removed from G in such way that G remains connected but it has no Kekulé structures. In this paper it is proved that the Anti-Kekulé number of all fullerenes is either 3 or 4 and that for each leapfrog fullerene the Anti-Kekulé number can be established by observing finite number of cases not depending on the size of the fullerene.     

The spread of unicyclic graphs with given size of maximum matchings

The spread s(G) of a graph G is defined as s(G) = max _i,j |λ _i − λ _j |, where the maximum is taken over all pairs of eigenvalues of G. Let U(n,k) denote the set of all unicyclic graphs on n vertices with a maximum matching of cardinality k, and U ^*(n,k) the set of triangle-free graphs in U(n,k). In this paper, we determine the graphs with the largest and second largest spectral radius in U ^*(n,k), and the graph with the largest spread in U(n,k).      

On Extremal Unicyclic Molecular Graphs with Prescribed Girth and Minimal Hosoya Index

Let G be an n-vertex unicyclic molecular graph and Z(G) be its Hosoya index, let F _n be the nth Fibonacci number. It is proved in this paper that if G has girth l then Z(G) ≥ F _l+1+(n−l)F _l +F _l-1, with the equality holding if and only if G is isomorphic to , the unicyclic graph obtained by pasting the unique non-1-valent vertex of the complete bipartite graph K _1,n-l to a vertex of an l-vertex cycle C _l . A direct consequence of this observation is that the minimum Hosoya index of n-vertex unicyclic graphs is 2n−2 and the unique extremal unicyclic graph is. The second minimal Hosoya index and the corresponding extremal unicyclic graphs are also determined.     

Hexagonal resonance of (3,6)-fullerenes

A (3,6)-fullerene G is a plane cubic graph whose faces are only triangles and hexagons. It follows from Euler’s formula that the number of triangles is four. A face of G is called resonant if its boundary is an alternating cycle with respect to some perfect matching of G. In this paper, we show that every hexagon of a (3,6)-fullerene G with connectivity 3 is resonant except for one graph, and there exist a pair of disjoint hexagons in G that are not mutually resonant except for two trivial graphs without disjoint hexagons. For any (3,6)-fullerene with connectivity 2, we show that it is composed of n(n ≥ 1) concentric layers of hexagons, capped on each end by a cap formed by two adjacent triangles, and none of its hexagons is resonant.     

On the Randić Index

The Randić index of an organic molecule whose molecular graph G is defined as the sum of (d(u)d(v))^−1/2 over all pairs of adjacent vertices of G, where d(u) is the degree of the vertex u in G. In Delorme et al., Discrete Math. 257 (2002) 29, Delorme et al gave a best-possible lower bound on the Randić index of a triangle-free graph G with given minimum degree δ(G). In the paper, we first point out a mistake in the proof of their result (Theorem 2 of Delorme et al., Discrete Math. 257 (2002) 29), and then we will show that the result holds when δ(G)≥ 2.     

On the Randić Index of Quasi-tree Graphs

The Randić index of an organic molecule whose molecular graph is G is the sum of the weights (d(u)d(v))^−1/2 of all edges uv of G, where d(u) and d(v) are the degrees of the vertex u and v in G. A graph G is called quasi-tree, if there exists such that G − u is a tree. In the paper, we give sharp lower and upper bounds on the Randić index of quasi-tree graphs. Mei Lu: Partially supported by NSFC (No. 10571105).     

Radical scavenging efficiency of different fullerenes C60-C70 and fullerene soot

This investigation was undertaken to determine the antioxidant activity of a range of fullerenes C₆₀ and C₇₀ in order to rank them according to their comparative efficiency. The model reaction of initiated (2,2′- azobisisobutyronitrile, AIBN) cumene oxidation was used to determine rate constants for addition of radicals to fullerenes. Measurements of oxidation rates in the presence of different fullerenes showed that the antioxidant activity as well as the mechanism and mode of inhibition were different for fullerenes C₆₀ and C₇₀ and fullerene soot. All fullerenes - C₆₀ of gold grade, C₆₀/C₇₀ (93/7, mix 1), C₆₀/C₇₀ (80 ± 5/20 ± 5, mix 2) and C₇₀ operated as alkyl radical acceptora, whereas fullerene soot surprisingly retarded the model reaction by a dual mode similar to that for the fullerenes and with an induction period like many of the sterically hindered phenolic and amine antioxidants. For the C₆₀ and C₇₀ the oxidation rates were found to depend linearly on the reciprocal square root of the concentration over a sufficiently wide range thereby fitting the mechanism for the addition of cumylalkyl radicals to the fullerene core. This is consistent with literature data on the more ready and rapid addition of alkyl and alkoxy radicals to the fullerenes compared with peroxy radicals. Rate constants for the addition of cumyl radicals to the fullerenes were determined to be k_(333K) = (1.9 ± 0.2) × 10⁸ (C₆₀); (2.3 ± 0.2) × 10⁸ (C₆₀/C₇₀, mix 1); (2.7 ± 0.2) × 10⁸ (C₆₀/C₇₀, mix 2); (3.0 ± 0.3) × 10⁸ (C₇₀), M⁻¹ s⁻¹. The increasing C₇₀ constituent in the fullerenes leads to a corresponding increase in the rate constant.The fullerene soot inhibits the model reaction according to the mechanism of trapping of peroxy radicals; the oxidation proceeds with a pronounced induction period and kinetic curves are linear in semi-logarithmic coordinates.For the first time the effective concentration of inhibiting centres and inhibition rate constants for the fullerene soot have been determined to be fn[C₆₀−soot] = (2.0 ± 0.1) × 10⁻⁴ mol g⁻¹ and k_inh = (6.5 ± 1.5) × 10³ M⁻¹ s⁻¹ respectively.The kinetic data obtained specify the level of antioxidant activity for the commercial fullerenes and scope for their rational use in different composites. The results may be helpful for designing an optimal profile of composites containing fullerenes.     

On lower bounds of number of perfect matchings in fullerene graphs

Couting perfect matchings in graphs is a very difficult problem. Some recently developed decomposition techniques allowed us to estimate the lower bound of the number of perfect matchings in certain classes of graphs. By applying these techniques, it will be shown that every fullerene graph with p vertices contains at least p/2+1 perfect matchings. It is a significant improvement over a previously published estimate, which claimed at least three perfect matchings in every fullerene graph. As an interesting chemical consequence, it is noted that every bisubstituted derivative of a fullerene still permits a Kekulé structure.     

On Unicycle Graphs with Maximum and Minimum Zeroth-order Genenal Randić Index

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as R _α⁰(G) = ∑ _v∈V(G) d _v ^α where α is an arbitrary real number. In this paper, we obtained the lower and upper bounds for the zeroth-order general Randić index R _α⁰(G) among all unicycle graphs G of order n. We give a clear picture for R _α⁰(G) of unicycle graphs according to real number α in different intervals.     

The Merrifield–Simmons index in (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">n</Emphasis> + 1)-graphs

A (n, n + 1)-graph G is a connected simple graph with n vertices and n + 1 edges. In this paper, we determine the upper bound for the Merrifield–Simmons index in (n, n + 1)–graphs in terms of the order n, and characterize the (n, n + 1)–graph with the largest Merrifield–Simmons index.     

Self-assembling of fullerene C<Subscript>60</Subscript> into (C<Subscript>60</Subscript>)<Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> clasters in aromatic solvents

Self-assembling of fullerene C₆₀ into (C₆₀) _n clusters in aromatic solvents was studied. The role of the π-π interactions and dispersion forces in the (C₆₀) _n cluster formation in these media is demonstrated using the data on the solubility of fullerene C₆₀ in these solvents and their ionization potentials and also spectral characteristics of fullerene C₆₀ in the range of 326–340 nm in different solvents.     

More Icosahedral Fulleroids

The discovery of the famous fullerene has raised an interest in the study of other candidates for a modeling of carbon molecules. Motivated by a P. Fowler's question Delgado Friedrichs and Deza defined I(a,b)-fulleroids as cubic convex polyhedra having only a-gonal and b-gonal faces and the symmetry groups isomorphic with the rotation group of the regular icosahedron. In this note we prove that for every n8 there exist infinitely many I(5,n)-fulleroids. This answers positively questions posed recently by Delgado Friedrichs and Deza.